- #1

MathematicalPhysicist

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what are they ?

i know they are related to quantum theory.

i know they are related to quantum theory.

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- Thread starter MathematicalPhysicist
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- #1

MathematicalPhysicist

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what are they ?

i know they are related to quantum theory.

i know they are related to quantum theory.

- #2

PRodQuanta

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Why explain shortly and possibly misinterpret when YOU can read?

Here ya go: http://arxiv.org/PS_cache/quant-ph/pdf/0101/0101012.pdf [Broken]

Enjoy

Paden Roder

Here ya go: http://arxiv.org/PS_cache/quant-ph/pdf/0101/0101012.pdf [Broken]

Enjoy

Paden Roder

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- #3

marcus

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Originally posted by PRodQuanta

Why explain shortly and possibly misinterpret when YOU can read?

Here ya go: http://arxiv.org/PS_cache/quant-ph/pdf/0101/0101012.pdf [Broken]

Enjoy

Paden Roder

I wouldn't always want to start downloading a PDF file from arxiv without first

looking at the abstract. Some articles have hundreds of pages.

And the title and brief summary can sometimes tell you enough. Here is the abstract for what Paden recommends reading. If you like the short summary in the abstract then click on "PDF" button right below it.

http://arxiv.org/quant-ph/0101012 [Broken]

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- #4

marcus

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http://arxiv.org/quant-ph/0101012 [Broken]

I'm impressed. Thanks for the link. It is Hardy's original article, only 34 pages, and gives the 5 axioms

Here is an exerpt from Hardy's article, "Quantum Theory from Five Reasonable Axioms"

This quote gives a taste of what it's like:

------------

[[[Definition:

The state associated with a particular preparation

is defined to be (that thing represented by) any

mathematical object that can be used to deter-

mine the probability associated with the out-

comes of any measurement that may be per-

formed on a system prepared by the given prepa-

ration.

Hence, a list of all probabilities pertaining to all pos-

sible measurements that could be made would cer-

tainly represent the state. However, this would most

likely over determine the state. Since most physical

theories have some structure, a smaller set of prob-

abilities pertaining to a set of carefully chosen mea-

surements may be sufficient to determine the state.

This is the case in classical probability theory and

quantum theory.

Central to the axioms are two inte-

gers K and N which characterize the type of system

being considered.

* The number of degrees of freedom, K, is defined

as the minimum number of probability measure-

ments needed to determine the state, or, more

roughly, as the number of real parameters re-

quired to specify the state.

* The dimension, N, is defined as the maximum

number of states that can be reliably distinguished from one another in a single shot measurement.

We will only consider the case where the number

of distinguishable states is finite or countably infinite. As will be shown below, classical probability theory has K = N and quantum probability theory has K = N

The five axioms for quantum theory (to be stated again, in context, later) are

Axiom 1 Probabilities. Relative frequencies (mea-

sured by taking the proportion of times a par-

ticular outcome is observed) tend to the same

value (which we call the probability) for any case

where a given measurement is performed on a

ensemble of n systems prepared by some given

preparation in the limit as n becomes infinite.

of N (i.e. K = K(N)) where N = 1; 2; : : : and

where, for each given N, K takes the minimum

value consistent with the axioms.

Axiom 2 Simplicity. K is determined by a function

of N (i.e. K = K(N)) where N = 1; 2; : : : and

where, for each given N, K takes the minimum

value consistent with the axioms.

Axiom 3 Subspaces. A system whose state is con-

strained to belong to an M dimensional subspace

(i.e. have support on only M of a set of N possi-

ble distinguishable states) behaves like a system

of dimension M.

Axiom 4 Composite systems. A composite system

consisting of subsystems A and B satisfies N =

N

Axiom 5 Continuity. There exists a continuous re-

versible transformation on a system between any

two pure states of that system.

The first four axioms are consistent with classical

probability theory but the fifth is not (unless the

word "continuous" is dropped). If the last axiom is

dropped then, because of the simplicity axiom, we

obtain classical probability theory (with K = N) in-

stead of quantum theory (with K = N

striking that we have here a set of axioms for quan-

tum theory which have the property that if a single

word is removed (namely the word "continuous" in

Axiom 5) then we obtain classical probability theory

instead.]]]

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- #5

marcus

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two core definitions and the fact that in a quantum

system the "degrees of freedom", as Hardy tells it, is equal

to the SQUARE of the dimension

(while in mere probability theory it is simply equal to the dimension itself) so I want to consider what he means by these two key numbers

Central to the axioms are two inte-

gers K and N which characterize the type of system

being considered.

* The number of degrees of freedom, K, is defined

as the minimum number of probability measure-

ments needed to determine the state, or, more

roughly, as the number of real parameters re-

quired to specify the state.

* The dimension, N, is defined as the maximum

number of states that can be reliably distinguished from one another in a single shot measurement.

We will only consider the case where the number

of distinguishable states is finite or countably infinite. As will be shown below, classical probability theory has K = N and quantum probability theory has K = N^{2}(note we do not assume that states are normalized).

In the quantum case, with K = N

"the minimum number of probability measurements needed to determine the state" is equal to the square of

"the maximum

number of states that can be reliably distinguished from one another in a single shot measurement"

Paden Roda, any comment about where the N-squared comes from?

- #6

PRodQuanta

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In the statement, axiom 5 states that there exists a continuous reversible transformation on a system between any two pure states of that system.

I think the term "reversible" means that it adds a whole new set of probabilities to the system. Thus, it would give the N it's square.

This may not make sense for 2 reasons:

1)I'm speaking in a language that I can understand, but maybe not descriptive enough for others.

2)Lack of knowledge on the subject.

There are my thoughts. Make of them what you will.

Paden Roder

- #7

MathematicalPhysicist

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i agree with you.Originally posted by marcus

I wouldn't always want to start downloading a PDF file from arxiv without first

looking at the abstract. Some articles have hundreds of pages.

And the title and brief summary can sometimes tell you enough. Here is the abstract for what Paden recommends reading. If you like the short summary in the abstract then click on "PDF" button right below it.

http://arxiv.org/quant-ph/0101012 [Broken]

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